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Properties

Label 2-1280-40.29-c1-0-13
Degree $2$
Conductor $1280$
Sign $0.998 - 0.0613i$
Analytic cond. $10.2208$
Root an. cond. $3.19700$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.806·3-s + (−1.67 − 1.48i)5-s + 2.15i·7-s − 2.35·9-s + 0.387i·11-s + 0.962·13-s + (1.35 + 1.19i)15-s − 1.61i·17-s − 6.31i·19-s − 1.73i·21-s + 6.15i·23-s + (0.612 + 4.96i)25-s + 4.31·27-s + 2i·29-s + 9.92·31-s + ⋯
L(s)  = 1  − 0.465·3-s + (−0.749 − 0.662i)5-s + 0.815i·7-s − 0.783·9-s + 0.116i·11-s + 0.266·13-s + (0.348 + 0.308i)15-s − 0.390i·17-s − 1.44i·19-s − 0.379i·21-s + 1.28i·23-s + (0.122 + 0.992i)25-s + 0.829·27-s + 0.371i·29-s + 1.78·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0613i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.998 - 0.0613i$
Analytic conductor: \(10.2208\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1280,\ (\ :1/2),\ 0.998 - 0.0613i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.006304465\)
\(L(\frac12)\) \(\approx\) \(1.006304465\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (1.67 + 1.48i)T \)
good3 \( 1 + 0.806T + 3T^{2} \)
7 \( 1 - 2.15iT - 7T^{2} \)
11 \( 1 - 0.387iT - 11T^{2} \)
13 \( 1 - 0.962T + 13T^{2} \)
17 \( 1 + 1.61iT - 17T^{2} \)
19 \( 1 + 6.31iT - 19T^{2} \)
23 \( 1 - 6.15iT - 23T^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 - 9.92T + 31T^{2} \)
37 \( 1 + 6.57T + 37T^{2} \)
41 \( 1 + 4.57T + 41T^{2} \)
43 \( 1 - 11.5T + 43T^{2} \)
47 \( 1 - 4.54iT - 47T^{2} \)
53 \( 1 - 8.96T + 53T^{2} \)
59 \( 1 + 6.31iT - 59T^{2} \)
61 \( 1 + 0.261iT - 61T^{2} \)
67 \( 1 - 9.89T + 67T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 + 13.0iT - 73T^{2} \)
79 \( 1 - 1.92T + 79T^{2} \)
83 \( 1 + 2.88T + 83T^{2} \)
89 \( 1 - 10.6T + 89T^{2} \)
97 \( 1 + 9.61iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.405980725868184040628893551793, −8.886693784317268080145180674599, −8.189247447154338072886005032824, −7.24376410995406441994150246875, −6.27700603074606369686103702902, −5.30290671650458056642234061984, −4.82910208916911797533382162224, −3.54470193666902261843457172300, −2.49247831715094952035617669357, −0.78334245472925390776570457047, 0.71007615304432389182521980669, 2.54061862939238809045673206706, 3.66297614760632944865938457318, 4.33599694802990530306320177642, 5.60030405960145607113303711467, 6.41535646832545681142893853670, 7.09656059465360012497027942459, 8.184157150477691446402266199044, 8.491010668318912585275972754717, 10.01083357641462383777105341285

Graph of the $Z$-function along the critical line